Journal of Inequalities in Pure and
Applied Mathematics
IMPROVED INCLUSION-EXCLUSION INEQUALITIES FOR
SIMPLEX AND ORTHANT ARRANGEMENTS
volume 2, issue 2, article 18,
2001.
DANIEL Q. NAIMAN AND HENRY P. WYNN
Department of Mathematical Sciences
Johns Hopkins University
Baltimore, MD 21218.
EMail: daniel.naiman@jhu.edu
URL: http://mts.jhu.edu/ dan
Department of Statistics
Warwick University
Coventry CV4 7AL, UK
EMail: hpw@stats.warwick.ac.uk
Received 6 September, 2000;
accepted 25 January, 2001.
Communicated by: S.S. Dragomir
Abstract
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
031-00
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Abstract
Improved inclusion-exclusion inequalities for unions of sets are available wherein
terms usually included in the alternating sum formula can be left out. This is
the case when a key abstract tube condition, can be shown to hold. Since the
abstract tube concept was introduced and refined by the authors, several examples have been identified, and key properties of abstract tubes have been described. In particular, associated with an abstract tube is an inclusion-exclusion
identity which can be truncated to give an inequality that is guaranteed to be at
least as sharp as the inequality obtained by truncating the classical inclusionexclusion identity.
We present an abstract tube corresponding to an orthant arrangement where
the inclusion-exclusion formula terms are obtained from the incidence structure
of the boundary of the union of orthants. Thus, the construction of the abstract
tube is similar to a construction for Euclidean balls using a Voronoi diagram.
However, the proof of the abstract tube property is a bit more subtle and involves consideration of abstract tubes for arrangements of simplicies, and intricate geometric arguments based on their Voronoi diagrams.
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
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2000 Mathematics Subject Classification: 52C99, 52B99, 60D05
Key words: Orthant arrangments, Inclusion-exclusion
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Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Voronoi Decomposition and Abstract Tube Based on Simplex Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 2 of 37
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3
Orthant Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
A Voronoi decomposition and abstract tube based
on orthant arrangements . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Properties of the Orthant Voronoi decomposition . . .
3.3
General position and dimension . . . . . . . . . . . . . . . . . .
3.4
Inclusion-Exclusion Inequalities and Identities for
Orthant Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Proofs of Propositions and Lemmas in Section 2 . . . . . . . . . .
5
Proofs of Propositions in Section 3 . . . . . . . . . . . . . . . . . . . . . .
References
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Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
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1.
Introduction
This paper continues work by the authors on a special class of indicator function and probability bounds of the inclusion-exclusion type [8, 9]. These are
are based on the abstract tube concept and give improvements over bounds produced by truncating the classical inclusion-exclusion identity.
Definition 1.1. An abstract tube is a finite collection of sets {A1 , . . . , An } and
a finite simplicial complex S with the following properties:
(i) every vertex of S corresponds to an index i ∈ {1, . . . , n}, so that S can be
viewed as a collection of subsets of {1, . . . , n}, and
S
(ii) T
whenever x ∈ ni=1 Ai the subsimplicial complex S(x) = {J ∈ S : x ∈
i∈J Ai } is contractible.
Definition 1.1 is slightly more general than the one in [9] in that we do not
require a one-to-one correspondence between vertices in the simplicial complex
S and the index set {1, . . . , n}. That is, the index set can be a superset of the
set of vertices. All of the properties of abstract tubes given in [9] remain valid
for this more general notion of abstract tube. In particular, associated with an
abstract tube is an inclusion-exclusion identity for ISni=1 Ai based on the terms
in S, which can be truncated to give an upper or lower bound. Furthermore,
abstract tubes with smaller simplicial complexes leade to sharper truncation
inequalities.
Since abstract tubes were introduced, there has been much interest by the
authors and others in uncovering new examples of them, while at the same
time, there has been reason to suspect that the interesting abstract tubes from
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
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a geometric point of view always arise from convex polyhedra. Certainly, for
the key examples appearing in [9] (see also [7]) where the sets Ai involved
are Euclidean balls or unions of half-spaces, a convex polyhedron is present, or
lurking, and plays a fundamental role in that it’s face incidence structure defines
the simplicial complex. Furthermore, the construction of these abstract tubes
always involves the nerve of a Voronoi diagram associated with the arrangement
of sets.
Dohmen [2, 3, 4, 5] has discovered some new classes of abstract tubes and
has demonstrated the utility of the abstract tube concept to network reliability.
While these classes of tubes provide many elegant examples with far-reaching
applications, the constructions tend to be graph-theoretic and the tubes are defined in combinatorial rather than geometric terms. Thus, they do not appear to
shed light on the question as to the generality of the Voronoi construction since
they apparently correspond to a different class of abstract tubes than the ones
considered in [9]. In fact, the authors have not been able to show that the abstract tube formed using balls and the associated Delauney simplicial complex
can be realized as one Dohmen’s class of abstract tubes.
In this paper, we address the above-mentioned question by describing a pair
of new and related examples of abstract tubes, associated with simplex arrangements and orthant arrangements, based on the Voronoi-type construction. The
abstract tube property for simplex arrangements is used to derive the abstract
tube property for orthant arrangments. While these examples are geometric, the
connection with polyhedra is considerably more complex, and the proof of the
abstract tube property uses a somewhat more intricate geometric argument than
in [9]. There remains the open question as to whether this more general proof
technique can be used to verify the abstract tube property for other examples.
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
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In Section 2, we develop the tools needed to give the abstract tube associated
with arrangements of simplices. The results of this section are key ingredients
in Section 3 where we treat abstract tubes based on orthant arrangements.
Aside from being of intrinsic geometric interest the abstract tube for orthants
can be used to derive improved reliability bounds for coherent systems. This
idea is developed in [10] and used there, in particular, to give a new inclusionexclusion identity for a k out of n system.
Improved Inclusion-Exclusion
Inequalities for Simplex and
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2.
Voronoi Decomposition and Abstract Tube Based
on Simplex Arrangements
The results of this section concern arrangements consisting of copies of a regular simplex in Rd , that is, translates of dilations of a simplex, and a certain
related Voronoi-type diagram. Simplex arrangements are closely related to arrangements consisting of translates of a single orthant in Rd+1 . In fact, the former is obtained by slicing the latter, and this point of view is very important
for what follows. It is also the case that, analogous to a certain construction for
balls (see [6]) properties of the Voronoi diagram are obtained by projecting the
boundary of the orthant arrangement onto the slicing subspace.
For convenience, because of the connection with orthant arrangements, we
identify Rd with the hyperplane
(
)
n
X
H = x ∈ Rd+1 :
xi = 0 ,
i=1
and we let πH : Rd+1 → H denote the linearPprojection onto this hyperplane,
d+1
1
so that πH (y) = y − y1, where y = d+1
i=1 yi , and 1 denotes the vector whose coordinates are all equal to 1. Let e(1) , . . . , e(d+1) denote the usual
orthonormal basis for Rd+1 . In order to simplify theqnotation below, we let
Pd+1 (i)
1
1
d
e = d+1
= d+1
1, and let ωd =k e − e(i) k= d+1
. Let
i=1 e
u(i) =
−πH (e(i) )
= ωd−1 (e − e(i) ), for i = 1, . . . , d + 1,
k πH (e(i) ) k
Improved Inclusion-Exclusion
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so that
(i)
(j)
hu , u i =
−1/d
1
if i 6= j
if i = j.
Having established a coordinate system for Rd+1 we can introduce the notation x x∗ for points x, x∗ ∈ Rd+1 to mean that xi ≤ x∗i for all i = 1, . . . , d+1,
and we use x ≺ x∗ to mean that all of the inequalities are strict. We also use the
notation x x∗ and x x∗ with the obvious reverse interpretation.
Each point y ∈ Rd+1 defines a closed orthant
Oy = x ∈ Rd+1 : x y
which is a translation y + O0 , of the usual nonnegative orthant. For b ∈ H and
r ≥ 0 define the regular d-simplex in H,
Ab,r =
d+1
\
Improved Inclusion-Exclusion
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D.Q. Naiman and H.P. Wynn
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(i)
(i)
x ∈ H : hx, u i ≤ hb, u i + r .
i=1
It is easy to see that Ab,r is the convex hull of the points b−rdu(i) , i = 1, . . . , d+
1. This simplex has barycenter b, the Euclidean distance from b to any of the
bounding hyperplanes of Ab,r is r, and the Euclidean distance from b to any
vertex is rd.
More generally, we allow r < 0 and still refer to the simplex Ab,r corresponding to the ordered pair (b, r). This level of generality, where we allow for
virtual simplices, is very important for the main result of the next section. Thus,
the notation Ab,r has a dual meaning as it can represent a set (possibly empty)
or an ordered pair. It will be clear from the context below which interpretation
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is appropriate. Generally speaking, when we use Ab,r to define a distance, we
use the pair (b, r). On the other hand, when we consider Boolean operations
involving simplices, then we use the notion of Ab,r as a set.
We will use the term arrangement of orthants in Rd+1 to mean a finite collection {Oy(i) , i = 1, . . . , n}, where y (i) are distinct elements of Rd+1 (Figure 2) and the term arrangement of simplices in Rd to mean a finite collection
{Ab(i) ,r(i) , i = 1, . . . , n}, where b(i) ∈ H and r(i) ∈ R and the pairs (b(i) , r(i) )
are distinct. Note that simplices in an arrangement are allowed to be empty
when viewed as sets. Figure 2 shows an orthant arrangement.
We introduce the distance to a simplex in Rd (H) by defining
dAb,r (x) =
max hx − b, u(i) i − r, for x ∈ H.
i=1,...,d+1
Observe that the simplex distance dAb,r (x) is negative, zero, or positive depending on whether x lies in the interior, the boundary or the complement of the
simplex Ab,r . If r < 0 then the distance is always negative, which is consistent
with the fact that as a set Ab,r is empty.
We use this simplex distance to associate a Voronoi-type diagram in H with
any arrangement of simplices in H. Given an arrangement {Ai = Ab(i) ,r(i) , i =
1, . . . , n} of simplices in H, (we allow for r(i) ≤ 0) we define
S(i|j) = x ∈ H : dAi (x) ≤ dAj (x) ,
and
Vi =
n
\
j=1
S(i|j) =
x ∈ H : dAi (x) = min dAj (x) .
j=1,...,d
Improved Inclusion-Exclusion
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Figure 1: An orthant arrangement. The vertices of the orthants are the points
where dotted line segments meet, and the solid line segments show where the
orthants share common boundaries.
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An important tool for constructing a simplicial complex from a collection of
sets is the nerve construction.
Definition 2.1. The nerve corresponding to a collection of sets {Vi , i = 1, . . . , n}
is
T the simplicial complex consisting of all index sets J ⊆ {1, . . . , n} for which
i∈J Vi 6= ∅.
The following
theorem, due to Borsuk [1], gives a topological connection
Sn
between i=1 Vi and the nerve of the collection {Vi , i = 1, . . . , n}.
Theorem 2.1. Given a collectionTof polyhedra {Vi , i = 1, . . . , n} in Rd with
the property that the intersection
i∈J Vi is either empty or contractible for all
Sn
J ⊆ {1, . . . , n}, the set i=1 Vi and a geometric realization of the nerve of
{Vi , i = 1, . . . , n} have the same homotopy type.
Now we can state the main result of this section.
Theorem 2.2. Given a simplex arrangement {Ab(i) ,r(i) , i = 1, . . . , n} let S be
the nerve of the corresponding Voronoi sets. Then the pair ({Ab(i) ,r(i) , i =
1, . . . , n}, S) forms an abstract tube.
The proof of this theorem requires several preliminary geometric propositions and lemmas, which we present first. The proofs of these may be found
in Section 4. For the remainder of this section we fix a simplex arrangement
{Ab(i) ,r(i) , i = 1, . . . , n} with Voronoi sets V1 , . . . , Vn as described above.
Proposition 2.3. Given a point y ∈ Rd+1 with y ≤ 0, we have Oy ∩ H = Ab,r
where b = y − y1 and r = −y/ωd .
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We refer to the simplex in Proposition 2.3 as the simplex corresponding to
the orthant Oy . More generally, we allow for y > 0 and we can still refer to the
simplex Ab,r , as the simplex corresponding to the orthant Oy , if b = y − y1 and
r = −y/ωd . Also, we can invert this operation and find a unique orthant Oy
corresponding to any given simplex Ab,r by taking y = b − rωd 1. This orthant
has the property that Ab,r = Oy ∩ H, if r ≥ 0. This construction also allows us
to associate an orthant arrangement in Rd+1 with any arrangement of simplices
in Rd , and vice versa. Figure 2 gives the simplex arrangement obtained by slicing the orthant arrangement in Figure 2 with the hyperplane H.
In addition, a ball (with respect to this distance) about a simplex is a simplex.
In fact, it is easy to see that
x ∈ H : dAb,r (x) ≤ s = Ab,r+s
Improved Inclusion-Exclusion
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as subsets of H.
T
T
Proposition 2.4. If ki=1 Ab(i) ,r(i) 6= ∅ then ki=1 Ab(i) ,r(i) = Ab,r where b =
−ωd (c − c1), r = −c, and where c ∈ Rd+1 has coordinates
cp = min hb(i) , u(p) i + r(i) , for p = 1, . . . , d + 1.
i=1,...,k
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In addition, maxi=1,...,k dAb(i) ,r(i) = dAb,r .
Observe that for a given point b ∈ H, the polyhedral cones
)
(
X
(k)
Cb = b −
λq u(q) : λq ≥ 0 ⊆ H,
q6=k
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Inequalities for Simplex and
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Figure 2: Simplex arrangement obtained by slicing the orthant arrangement in
Figure 2 with the hyperplane H.
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(with vertex b) cover H and meet only on their (relative) boundaries, which we
S
(k)
(k)
(k)
denote by ∂Cb , so that a point x ∈ H\ d+1
lies in the interior of Cb
k=1 ∂Cb
for a unique choice of index k. We express the simplex distance for a point in
one of these cones in the following.
P
Proposition 2.5. Given b ∈ H and r ≥ 0 and a point x = b − q6=k λq u(q) ∈
P
(k)
Cb , we have dAb,r (x) = d1 q6=k λq − r.
o
n
Proposition 2.6. The set s ∈ R : x + s1 ∈ Ob−rωd 1 forms an interval
[ωd dAb,r (x), +∞), for all r ∈ R and b, x ∈ H.
Let y (i) = b(i) − r(i) ωd 1 so that the orthant Oi = Oy(i) corresponds to Ai . As
an immediate consequence of Proposition 2.6, we see that
)
(
n
n
[
[
s ∈ R : x + s1 ∈
Oi = [ωd dAi (x), +∞) = [ωd min dAi (x), +∞).
i=1
i=1,...,n
i=1
Sn
for any point x ∈ H. Thus, the map Ψ : H → ∂ { i=1 Oi } taking
S x to
x+ωd mini=1,...,n dAi (x)1 gives a homeomorphism between H S
and ∂ { ni=1 Oi }
whose inverse is the restriction of the projection map πH to ∂ { ni=1 Oi } . Using
Proposition 2.6, it follows that
!int
n
[
Ψ(Vi ) = Oi \
Oj
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j=1
The following two Lemmas form a crucial step in establishing the abstract
tube property below. It ensures that Borsuk’s Theorem 2.1 can be applied to
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Figure 3: The Voronoi diagram associated with the simplex arrangement in
Figure 2. Observe that the boundaries of the Voronoi sets correspond to the
dashed line segments in Figure 2 and the solid lines in Figure 2.
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equate the homotopy type of the union of a collection of Voronoi sets with the
nerve of the collection of Voronoi sets. These same results were essential in
proving the abstact tube property for balls appearing in [8].
T
Lemma 2.7. For every J ⊆ {1, . . . , n} the intersection i∈J Vi is either empty
or contractible.
Lemma 2.8. If J ⊆ {1, . . . , n} then
[
[\
Vi =
S(i|j).
i∈J
Improved Inclusion-Exclusion
Inequalities for Simplex and
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i∈J j ∈J
/
D.Q. Naiman and H.P. Wynn
The following result, which is specific to simplex arrangements and their
Voronoi diagrams, gives a crucial geometric observation leading to the proof of
Theorem 2.2.
S
T
Lemma 2.9. If x∗ ∈ ni=1 Ai and J = {i : x∗ ∈ Ai } thenT i∈I S(i|j) is
nonempty and star-shaped with respect to the barycenter b of i∈I Ai , for all
I ⊆ J and j ∈
/ J.
Figure 4 illustrates the star-shaped property in Lemma 2.9.
S
Proof of Theorem 2.2. Fix x∗ ∈ T ni=1 Ai . We must show the subsimplicial
complex S(x∗ ) = {I ∈ S : x∗ ∈ i∈I Ai } is contractible. Let J = {i : x∗ ∈
Ai } so that S(x∗ ) is the nerve of the collection {Vi , i ∈ J}. By S
Lemma 2.7
and Borsuk’s Theorem 2.1, S(x∗ ) has the same homotopy type as i∈J Vi . By
Lemma 2.8, we can write
[
[
Vi =
Ti ,
i∈J
i∈J
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x2
D.Q. Naiman and H.P. Wynn
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x1
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Figure 4: Illustration of the star-shaped property in Lemma 2.9. There are 4
triangles with centers labeled x1 , . . . , x4 . The triangles
T centered at x1 , x2 and x3
intersect to form another triangle T, and the set i=1,2,3 S(i|4), is star-shaped
with respect to the barycenter of T. The boundaries of the regions S(i|4), i =
1, 2, 3 are drawn using dotted lines.
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where
Ti =
\
S(i|j), for i ∈ J.
j ∈J
/
T
If I ⊆ J andT
we write i∈I Ai = Ab,r as in Proposition 2.4, then Lemma 2.9
guarantees that i∈I S(i|j) is star-shaped with respect to the barycenter b for all
j∈
/ J. It follows that
\
\\
\\
S(i|j)
Ti =
S(i|j) =
i∈I
i∈I j ∈J
/
j ∈J
/ i∈I
is also star-shaped with respect to b. Since every such intersection is star-shaped,
S
and hence contractible, Borsuk’s Theorem 2.1 allows us to conclude that i∈J Tj
has the same homotopy
type as the nerve of the collection {Tj , j ∈ J}. But evT
ery intersection i∈I Ti is nonempty, so the nerve forms a simplex, which is
contractible.
Remark 2.1. It is of interest to compare the proof of the abstract tube property
with the proof appearing in [8] for the case of balls of equal radius, when the
nerve of the usual Voronoi diagram is used to form the simplicial complex.
There, contractibility of the subsimplicial
complex S(x∗ ) follows from the fact
S
that the union of Voronoi sets i∈J Vi is star-shaped with respect to x∗ . In the
present case, we do not in general have star-shapedness of this set, but we are
able to prove contractibility by representing this union as a union of pieces
which always intersect in nonempty star-shaped pieces.
Remark 2.2. Since the distance to a simplex Ab,r satisfies dAb,r+c (x) = dAb,r (x)+
c, it follows that the Voronoi decomposition of H (and hence the associated
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simplicial complex S) corresponding to a simplex arrangement {Ab(i) ,r(i) , i =
1, . . . , n} is unaffected if we add the same constant to each r(i) .
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3.
Orthant Arrangements
3.1.
A Voronoi decomposition and abstract tube based on orthant arrangements
Now we apply the results of the previous section to give an analogous result
for arrangements consisting of translates of the orthants. To keep the notation
consistent with that of the last section, we consider translates of the negative
orthant in Rd+1 . We first introduce an orthant distance, which measures the
distance to an orthant Oy . Let
d˜Oy (x) =
D.Q. Naiman and H.P. Wynn
max {yj − xj }.
j=1,...,d+1
Observe that d˜Oy (x) is less than, equal to, or greater than 0 respectively, depending on whether x lies in the interior, boundary or exterior of Oy .
A collection of orthants {Oy(i) , i = 1, . . . , n, } where the y (i) are distinct, will
be referred to as an orthant arrangement in Rd+1 . Given such an arrangement,
the orthant distance is used to define a Voronoi decomposition of Rd+1 by letting
Ṽi =
n
\
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S̃(i|j)
j=1
where
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
n
o
S̃(i|j) = x ∈ Rd+1 : d˜Oy(i) (x) ≤ d˜Oy(j) (x) .
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The main result of this section is the following.
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Theorem 3.1. If {Oy(i) , i = 1, . . . , n} is an orthant arrangement in Rd+1 , then
the pair ({Oy(i) , i = 1, . . . , n}, S̃) forms an abstract tube, where S̃ denotes the
nerve of the corresponding Voronoi decomposition {Ṽi , i = 1, . . . , n} of Rd+1 .
Some preliminary Propositions will play a key role in the proof of Theorem 3.1. Proofs of the results presented in this section, except for the proof of
the main result, Theorem 3.1, appear in Section 5.
Proposition 3.2. Given an orthant arrangement {Oy(i) , i = 1, . . . , n}, the nerve
of the corresponding Voronoi decomposition {Ṽi , i = 1, . . . , n} coincides with
the nerve of {Ṽi ∩ H, i = 1, . . . , n}.
The Voronoi decomposition for orthants is closely related to the one in the
last section, and exploiting this connection is the key to proving the main result
of this section. The basic idea is to introduce a simplex arrangement associated
with a given orthant arrangement (as in the remark following Propositon 2.3)
{Oy(i) , i = 1, . . . , n},
by taking
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
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{Ab(i) ,r(i) , i = 1, . . . , n},
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(i)
= −y /ωd .
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Proposition 3.3. Given any orthant arrangement {Oy(i) , i = 1, . . . , n}, in Rd+1 ,
let {Vi , i = 1, . . . , n} be the Voronoi decomposition for the associated simplex
arrangement. Then the Voronoi decomposition for the orthant and simplex arrangements are related in that
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where b
(i)
=y
(i)
(i)
− y 1 and r
(i)
Ṽi ∩ H = Vi ,
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and consequently the nerves of the decompositions coincide.
Finally, we will need the following.
Proposition 3.4. Given x, y ∈ Rd+1 we have x ∈ Oy if and only if x − x1 ∈
Ay−y1,(x−y)/ωd .
S
Proof of Theorem 3.1. Fix x ∈ ni=1 Oy(i) and let J˜ = {i : x ∈ Oy(i) }. We
need to show that the subsimplicial complex defined by
\
˜
S̃(x) = {I ∈ S̃ : x ∈
Oy(i) } = {I ∈ S̃ : I ⊆ J}
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
i∈I
is contractible.
Consider the simplex arrangement obtained by applying the same construction in Proposition 3.4 to each of the orthants Oy(i) , that is, take {Ab(i) ,r(i) , i =
1, . . . , n}, where b(i) = y (i) − y (i) 1, and r(i) = (x − y (i) )/ωd . Let {Vi , i =
1, . . . , n} be the Voronoi decomposition for this simplex arrangment, and let S
denote the corresponding nerve. This Voronoi decomposition is unchanged if
we subtract the same constant (x/ωd ) from all of the r(i) , but this modification
leads to the simplex arrangement associated with the original orthant arrangement. We conclude that {Vi , i = 1, . . . , n} is also the Voronoi diagram for this
simplex arrangement. By Proposition 3.3 we conclude that S = S̃.
By Theorem 2.2 ({Ab(i) ,r(i) , i = 1, . . . , n}, S) forms an abstract tube so if we
let J = {i : x − x1 ∈ Ab(i) ,r(i) } then the subsimplicial complex defined by
\
S(x − x1) = {I ∈ S : x − x1 ∈
Ab(i) ,r(i) } = {I ∈ S : I ⊆ J}
i∈I
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is contractible. But Proposition 3.4 guarantees that J = J˜ so using the fact that
S̃ = S we can conclude that
S̃(x) = S(x − x1)
so S̃(x) is contractible.
3.2.
Properties of the Orthant Voronoi decomposition
The Voronoi decomposition {Ṽi , i = 1, . . . , n} corresponding to a given orthant
arrangement {Oy(i) , i = 1, . . . , n} has a simple description in terms of the decomposition of the boundary of the union of the orthants. This description helps
us in calculating the simplicial complex S̃.
Let
!int
n
[
Bi = ∂Oy(i) \
Oy(i)
i=1
so that the Bi define a decomposition of the boundary
!
n
n
[
[
B=∂
Oy(i) =
Bi .
i=1
i=1
Proposition 3.5. For a nonempty index set J, we have J ∈ S̃ if and only if
T
i∈J Bi 6= ∅. In other words, the nerve of the Voronoi decomposition {Ṽi , i =
1, . . . , n} coincides with the nerve of the decomposition {Bi , i = 1, . . . , n} of
B.
Improved Inclusion-Exclusion
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D.Q. Naiman and H.P. Wynn
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Definition 3.1. An orthant
y (i) in an orthant arrangement {Oy (i) , i = 1, . . . , n}
Sn Oint
is exposed if Oy(i) 6⊆ j=1 Oy(j) .
S
S
(i)
Observe that Oy(i) ⊆ nj=1 Oyint
∈ nj=1 Oyint
(j) if and only if y
(j) , and this is
(i)
(j)
in turn is equivalent to y y for some j 6= i. Thus, the exposed orthants
correspond to those indices i for which y (i) 6 y (j) for all j.
We use the notation maxj∈J y (j) to mean the coordinatewise maximum of
the y (j) for j ∈ J. As consequence of Proposition 3.5 we have the following
description of the faces of the nerve of the Voronoi decomposition.
Corollary 3.6. The faces of S̃ correspond to the (nonempty) index sets J for
which maxi∈J y (i) 6 y (j) for all j. In particular, the vertices of S̃ (the single
element faces) correspond to the exposed orthants.
Following Corollary 3.6 we can say equivalently that the index set J, or the
point y = maxi∈J y (i) , or the orthant Oy is covered.
3.3.
General position and dimension
For a generic orthant arrangement {Oy(i) , i = 1, . . . , n} in Rd+1 the simplicial
complex defining the tube above, that is, the nerve S̃ of the Voronoi decomposition, has dimension d + 1. As a consequence, the inclusion-exclusion identity
has depth d + 2 instead of n, which can lead to a dramatic improvement. We
make this rigorous as follows.
Definition 3.2. An orthant arrangement {Oy(i) , i = 1, . . . , n} in Rd+1 is in gen(i)
eral position if for every coordinate index j the values yj , i = 1, . . . , n are
(i)
(k)
distinct. In other words, yj = yj for some i, j, k implies i = k.
Improved Inclusion-Exclusion
Inequalities for Simplex and
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The orthant arrangements that fail to be in general position define a set of
Lebesgue measure zero in the set of orthant arrangements. Under the general
position assumption the dimension of the simplicial complex defining the tube
has the right dimension.
Proposition 3.7. If an orthant arrangement is in general position then the simplicial complex S̃ defining the abstract tube in Theorem 3.1 has dimension at
most d + 1.
When an orthant arrangement fails to be in general position, it is still possible
to perturb it slightly to attain general position, and use the modified arrangement
to obtain improved inclusion-exclusion identities and inequalities that are valid
almost everywhere. This idea is explored in [9] for abstract tubes related to
polyhedra, and an analogous result can be used in the present context. In [10],
abstract tubes based on orthant arrangements are used to derive new reliability
bounds for coherent systems, and in that context, perturbation is used to give
even further improved inclusion-exclusion indicator identities and inequalities.
3.4.
Inclusion-Exclusion Inequalities and Identities for Orthant Unions
Using Theorem 4 in [9] the abstract tube property leads immediately to the
following.
Theorem 3.8. Given a finite collection of distinct points y (i) , i = 1, . . . , n in
Rd , define
Improved Inclusion-Exclusion
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S = {J ⊆ {1, . . . , n} : max y (i) 6 y (j) , for all j = 1, . . . , n}.
i∈J
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Then the following indicator function inequalities hold
(−1)m+1 ISni=1 Oi
≤ (−1)m+1

m
X

X
(−1)k+1
k=1
ITj∈J Oj


, for m = 1, 2, . . . , D,

J∈S : |J|=m
where Oi denotes Oy(i) , and D = max{|J| : J ∈ S}. In addition, equality
holds for m = D. Each inequality is at least as sharp as the corresponding
classical inclusion-exclusion inequality


m
X

X
m+1 Sn
m+1
k+1
T
(−1)
I i=1 Oi ≤ (−1)
(−1)
I j∈J Oj ,


k=1
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
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J⊆{1,...,n} : |J|=m
Contents
corresponding to the abstract tube using a simplicial complex composed of all
nonempty index sets.
The theorem also holds if we use negative orthants instead of positive ones,
that is, if we use as the definition of Oy(i)
{x ∈ Rd : x y (i) },
and if we redefine S to be
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{J ⊆ {1, . . . , n} : min y (i) 6≺ y (j) , for all j = 1, . . . , n}.
i∈J
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4.
Proofs of Propositions and Lemmas in Section 2
Proof of Proposition 2.3 Here, we are viewing a simplex as a set. For any x ∈ H
we have x y if and only if
hx, −e(i) i ≤ hy, −e(i) i, for i = 1, . . . , d + 1.
Since hz, ei = 0 for z ∈ H, this is equivalent to
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
hx, e − e(i) i ≤ hy − y1, e − e(i) i − hy1, e(i) i for i = 1, . . . , d + 1,
D.Q. Naiman and H.P. Wynn
(i)
which, upon dividing by ωd =k e
− e k leads to the equivalent condition
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hx, u(i) i ≤ hy − y1, u(i) i − yh1, e(i) i/ωd for i = 1, . . . , d + 1.
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Proof of Proposition 2.4. For the first claim, we can use the comment following
Proposition 2.3 to write Ab(i) ,r(i) = Oy(i) ∩ H, where y (i) = b(i) − r(i) ωd 1. Then
we have
k
k
\
\
Ab(i) ,r(i) =
Oy(i) ∩ H = Oz ∩ H
i=1
i=1
where z = maxi=1,...,k y (i) , the maximum being coordinatewise. A straightforward calculation gives
zp = −ωd min hy (i) , u(p) i + r(i) ,
i=1,...,k
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so the result follows from the application of Propsition 2.3 For the second claim,
we have
max hx, u(p) i − hb(i) , u(p) i − r(i)
= max hx, u(p) i − min hb(i) , u(p) i + r(i)
p=1,...,d+1
i=1,...,k
= max hx, u(p) i − hb, u(p) i + r
max dAb(i) ,r(i) (x) = max
i=1,...,k
i=1,...,k p=1,...,d+1
p=1,...,d+1
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
= dAb,r (x).
Proof of Proposition 2.5. We have
)
(
)
(
X
X
λq u(q) , u(p) i − r = − min
λq hu(q) , u(p) i − r.
dAb,r (x) = max h−
p
q6=k
The result then follows from the fact that
1P
X
− d q6=k λq
(q)
(p)
P
λq hu , u i =
− d1 q6=p,k λq + λp
q6=k
p
q6=k
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if p = k
if p 6= k.
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Proof of Proposition 2.6. Since x + s1 ∈ Ob−rωd 1 if and only if x + s1 n
o
b − rωd 1, we see that the set s ∈ R : x + s1 ∈ Ob−rωd 1 , forms an interval
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that is closed on the left and extends to infinity on the right. The minimum value
of s in this interval is given by
max −(xi − bi ) − rωd =
i=1,...,d+1
=
=
max −hx − b, e(i) i − rωd
i=1,...,d+1
max −hx − b, e(i) − ei − rωd
i=1,...,d+1
max hx − b, ωd u(i) i − rωd
i=1,...,d+1
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
= ωd dAb,r (x).
Proof of Lemma 2.7. Since Ψ is a homeomorphism, and
!
!int
n
\
\
\
[
Ψ( Vi ) =
Ψ(Vi ) =
Oj \
Oi
i∈J
i∈J
i=1
j∈J
it suffices to show that if the set
!
W =
\
j∈J
Oj
\
n
[
!int
Oi
i=1
is nonempty, then it is contractible.
Suppose z ∈ W so that z y (j) for all j ∈ J and z 6Ty (i) , for i = 1, . . . , n.
If we define v = maxj∈J y (j) then observe that v ∈ j∈J Oj , and z v.
Furthermore, if it were the case that v ∈ Oiint for some index i, so that v y (i) ,
D.Q. Naiman and H.P. Wynn
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then we would have z y (i) and this is a contradiction. We conclude that
v ∈ W.
We proceed to show W is star-shaped with respect to v. Suppose w ∈ W
and λ ∈ [0, 1] then wλ = (1 − λ)w + λv ∈ Oj for all j ∈TJ by convexity.
We proceed to show wλ ∈
/ Oiint for i = 1, . . . , n. Since w ∈ j∈J Oj we have
w v, and it follows that w wλ v. Consequently, if wλ ∈ Oiint we obtain
w ∈ Oiint , which is a contradiction.
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
Proof of Lemma 2.8. On the one hand
[
i∈J
Vi =
n
[\
S(i|j) ⊆
i∈J j=1
[\
D.Q. Naiman and H.P. Wynn
S(i|j).
i∈J j ∈J
/
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S
T
∗
On the other hand, suppose x ∈ i∈J j ∈J
/ S(i|j) so that for some i ∈ J we
have dAi∗ (x) ≤ dAj , for all j ∈
/ J. Let i∗∗ ∈ J minimize
Ai∗∗ (x). It follows
S dT
that dAi∗∗ (x) ≤ dAj (x) for all j = 1, . . . , n, that is x ∈ i∈J nj=1 S(i|j).
T
Proof
of Lemma 2.9. We can use Proposition 2.4 to write i∈I Ai = Ab,r since
T
i∈I Ai 6= ∅, b being the barycenter of the simplex Ab,r . Using the second part
of Proposition 2.4, we see that
\
\
S(i|j) = {x : dAi (x) ≤ dAj (x)}
i∈I
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i∈I
= {x : max dAi (x) ≤ dAj (x)}
i∈I
= {x : dAb,r (x) ≤ dAj (x)}.
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T
To prove the claim of star-shapedness of i∈I S(i|j) it suffices to show
T that the
intersection of any ray emanating from the barycenter b withPthe set i∈I S(i|j)
forms a line segment containing b. So fix a ray, say {b − η q6=k λq u(q) : η ≥
0}, for some index k and nonnegative constants λq for q 6= k, and define
X
f (η) = dAb,r (b − η
λq u(q) ),
q6=k
and
g(η) = dAj (b − η
X
(q)
λq u ).
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
q6=k
The proof will be complete once we have demonstrated that
V = {η ≥ 0 : f (η) ≤ g(η)}
is an interval containing 0.
P
1
Using Proposition 2.5, we obtain f (η) = d q6=k λq η − r. Thus, we see
that
P
(i) f is linear, with f (0) = −r and slope d1 q6=k λq .
On the other hand, from the definition of simplex distance
g(η) = maxp=1,...,d+1 gp (η), where
X
gp (η) = hb − η
λq u(q) − b(j) , u(p) i − r(j) .
q6=k
Since
Ab,r =
d+1
\
p=1
{x ∈ H : hx, u(p) i ≤ hb, u(p) i + r}
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and
Aj =
d+1
\
{x ∈ H : hx, u(p) i ≤ hb(j) , u(p) i + r(j) }
p=1
and Ab,r 6⊆ Aj it must be the case that
hb, u(p) i + r > hb(j) , u(p) i + r(j)
for some index p. This leads to the conclusion that
(ii) g(0) = maxp=1,...,d+1 hb − b(j) , u(p) i − r(j) > −r = f (0)
Finally, each function gp is linear g is piecewise linear and convex.
P
In addition, the slope of gp is given by −h q6=k λq u(q) , u(p) i, so the same
calculation as in the proof of Proposition 2.5 shows that the maximum slope
occurs for gk , and this function has the same slope as f. We have therefore
shown that
(iii) g is piecewise linear and convex (and continuous), and the maximum slope
of g, where g is differentiable, is the same as the slope of f.
Using properties (i), (ii) and (iii), it is easy to see that 0 ∈ V, and either the
graphs of f and g do not cross, or they cross at a single point, or they meet
in an interval of the form [η ∗ , +∞). In each case, the set V forms an interval
containing 0.
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
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5.
Proofs of Propositions in Section 3
Proof of Proposition 3.2 The orthant distance satisfies
d˜Oy (x + c1) = d˜Oy (x) + c,
and consequently
d˜Oy(i) (x + c1) ≤ d˜Oy(j) (x + c1)
if and only if
d˜Oy(i) (x) ≤ d˜Oy(j) (x).
Thus, each Ṽi is the union of the set of lines of the form {x + c1 : c ∈ R}
where x ∈ Ṽi ∩H. It follows immediately that the nerve of the {Ṽi , i = 1, . . . , n}
coincides with the nerve of the {Ṽi ∩ H}.
Proof of Proposition 3.3 For x ∈ H a straightforward calculation shows that
dAb(i) ,r(i) (x) = max hx, u(p) i − hb(i) , u(p) i − r(i)
p
= max {yp − xp } /ωd
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Inequalities for Simplex and
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p
= d˜Oy(i) (x)/ωd .
Thus
dAb(i) ,r(i) (x) ≤ dAb(j) ,r(j) (x),
if and only if
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for x ∈ H. The second claim follows from Proposition 3.2.
Proof of Proposition 3.4 We have xp ∈ Oy if and only if x − x1 ∈ Oy−x1 ,
which is equivalent to
d˜O
(x − x1) ≤ 0.
y−x1
Since x − x1 ∈ H we can use the calculation in the proof of Proposition 3.3 to
conclude that an equivalent condition is
dA
y−y
1,(x−y)/ωd
(x − x1) ≤ 0,
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
and this gives the desired result.
T
T
∗
Proof of Proposition 3.5 If i∈J Ṽi 6= ∅, fix x ∈
=
i∈J Ṽi . Let d
minj=1,...,n dOy(j) (x), so that dOy(i) (x) = d∗ for i ∈ J and dOy(i) (x) > d∗ for
i∈
/ J. If x∗ = x + d∗ 1 then we have dOy(i) (x∗ ) = 0 for i ∈ J and dOy(i) (x∗ ) > 0
Sn
int Sn
for i ∈
/ J, thus x∗ ∈ ∂Oy(i) for i ∈ J, and x∗ ∈
/
Oy(i)
= i=1 Oyint
(i) .
i=1
T
T
∗
We conclude that x ∈ i∈J Bi . Conversely, if x ∈ i∈J Bi then for i ∈ J we
have x ∈ ∂Oy(i) so d˜Oy(i) (x) = 0. Furthermore, for all i we have x ∈
/ Oyint
(i) so
T
d˜O (x) ≥ 0. We conclude therefore, that x ∈
Ṽi .
y (i)
i∈J
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Proof of Corollary 3.6 We use the characterization of faces in Proposition 3.5.
Fix a nonempty index set J and let m = maxi∈J y (i) .
T
T
Suppose i∈J Bi 6= ∅, and let x ∈ i∈J Bi . Then x ∈ Oy(i) for i ∈ J and
(i)
x∈
/ Oyint
and hence x m. Furthermore, we
(i) for all i. It follows that x y
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cannot have m y (j) for some j since this would give x y (j) . This proves
that m satisfies the stated condition.
S
Conversely, if m 6 y (j) for all j, then for i ∈ J we have m ∈ Oy(i) \ ni=1 Oyint
(i)
T
= Bi , so i∈J Bi 6= ∅.
Proof of Proposition 3.7 Suppose an index set J defines a face in S, and let
m = maxi∈J y (i) . By the general position assumption, for each coordinate index
(i )
j there is a unique index ij ∈ J such that mj = yj j . If |J| ≥ d + 2 then since
{ij , j = 1, . . . , d + 1} consists of at most d + 1 elements, there must be some
index k ∈ J\{ij , j = 1, . . . , d + 1}. It follows that m y (k) , which contradicts
the characterization in Corollary 3.6.
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
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References
[1] BORSUK, On the imbedding of systems of compacta in simplicial complexes, Fund. Math., 35 (1948), 217–234.
[2] K. DOHMEN, Inclusion-exclusion and network reliability, Electron. J.
Combin., 5(1) (1998), Research Paper 36, 8 pp. (electronic).
[3] K. DOHMEN, An improvement of the inclusion-exclusion principle, Arch.
Math., (Basel), 72(4) (1999), 298–303.
[4] K. DOHMEN, Improved inclusion-exclusion identities and inequalities
based on a particular class of abstract tubes, Electron. J. Probab., 4(5)
(1999), 12 pp. (electronic).
Improved Inclusion-Exclusion
Inequalities for Simplex and
Orthant Arrangements
D.Q. Naiman and H.P. Wynn
Title Page
[5] K. DOHMEN, Convex geometric inclusion-exclusion identities and Bonferroni inequalities with applications to system reliability analysis and
reliability covering problems, Technical report: Informatik-Bericht, 132,
Humboldt-University Berlin, (1999).
[6] H. EDELSBRUNNER, The union of balls and its dual shape, Discrete and
Computational Geometry, 13 (1995), 415–440.
[7] H. EDELSBRUNNER, Algorithms in Combinatorial Geometry, SpringerVerlag, Heidelberg, Germany, (1987).
[8] D. NAIMAN AND H.P. WYNN, Inclusion-exclusion-Bonferroni identities
and inequalities for discrete tube-like problems via Euler characteristics,
Annals of Statistics, 20 (1992), 43–76.
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[9] D. NAIMAN AND H.P. WYNN, Abstract Tubes, Improved InclusionExclusion Identities and Inequalities, and Importance Sampling, Annals
of Statistics, 25 (1997), 1954–1983.
[10] D. NAIMAN AND H.P. WYNN, Reliability bounds for coherent systems:
an approach based on abstract tubes, manuscript in preparation, (2000).
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Inequalities for Simplex and
Orthant Arrangements
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